Nonlinear inelastic analysis of steel-concrete composite beam-columns using the stability functions

Transcription

1 Structural Egieerig ad Mechaics, Vol. 30, No. 6 (2008) Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios Jug-Woog Park ad Seug-Eock Kim Departmet of Civil ad Evirometal Egieerig, Sejog Uiversity, 98 Guja-dog, Gwagji-gu, Seoul, , Korea (Received August 26, 2008, Accepted November 10, 2008) Abstract. I this study, a flexibility-based fiite elemet method cosiderig geometric ad material oliearities is developed for aalyzig steel-cocrete frame structures. The stability fuctios obtaied from the exact bucklig solutio of the beam-colum subjected to ed momets are used to accurately capture the secod-order effects. The proposed method uses the force iterpolatio fuctios, icludig a momet magificatio due to the axial force ad lateral displacemet. Thus, oly oe elemet per a physical member ca accout for the iteractio betwee the bedig momet ad the axial force i a ratioal way. The proposed method applies the Newto method based o the load cotrol ad uses the secat stiffess method, which is computatioally both efficiet ad stable. Accordig to the evaluatio result of this study, the proposed method cosistetly well predicts the oliear ielastic behavior of steel-cocrete composite frames ad gives good efficiecy. Keywords: oliear aalysis; stability fuctios; beam-colum elemet; composite structures; cocretefilled steel tube. 1. Itroductio Recetly, researchers have focused attetio o force-based ad mixed formulatios that permit more accurate represetatio of the force distributio alog the elemet. Sice there are o displacemet iterpolatio fuctios to relate the sectio deformatios to odal displacemets, it is still challegig to implemet the flexibility-based methods for oliear problems i the cotext of a fiite elemet program. Ciampi ad Carlesimo (1986) proposed the cosistet formulatio of force-based elemets. Their procedure was improved by Taucer, Spacoe, ad Filippou (1991), ad Spacoe, Filippou, ad Taucer (1996) to develop a force-based fiber frame elemet for oliear aalysis of reiforced cocrete structures. Spacoe, Ciampi, ad Fillipou (1996) proposed the geeral formulatio of a mixed approach, which poits the way to the cosistet umerical implemetatio of the elemet state determiatio. Ayoub ad Filippou (2000) preseted a ielastic beam elemet for the aalysis of steel-cocrete girders with partial composite usig a two-field mixed formulatio with idepedet approximatio of iteral forces ad trasverse displacemets. Professor, Correspodig author, sekim@sejog.ac.kr

2 764 Jug-Woog Park ad Seug-Eock Kim Ayoub (2003) exteds the mixed formulatio to aalyze ielastic beams o foudatios, where both the beam ad foudatio are assumed to be ielastic. Alemdar ad White (2005) developed both a first order flexibility-based model withi a corotatioal approach ad a two-field mixed model. Hjelmstad ad Taciroglu (2002) compared the variatioal approaches with the so-called oliear flexibility methods that have recetly bee reported i the literature, ad coclude that these approaches, while certaily havig merit, are ot variatioally cosistet. Although the geometric secod-order effects are importat i the frame structures ad sometimes they are quite plai, most of the force-based methods do ot cosider the geometric secod-order effects. The structural aalysis icludig the secod-order effects is further complicated by the fact that the resultig equilibrium equatios are differetial equatios istead of the usual algebraic equatios. For a sleder colum, bucklig may occur whe all fibers of the cross-sectio are still elastic. Thus, the Euler load will gover the load-carryig capacity of the colum. For a short colum, material yieldig of the fibers i the cross-sectio usually occurs before bucklig takes place, so the yieldig force of the sectio will gover the limit state of the colum. I reality of practice, some of the fibers of the cross-sectio may yield while some fibers still remai elastic, ad the failure may be more accurately described as combied bucklig of the colum ad material yieldig of the sectio. I this study, a flexibility-based fiite elemet method cosiderig geometric secod-order effects ad material ielasticity is developed to improve the commo flexibility-based methods which have bee proposed up to ow. The stability fuctios obtaied from the exact bucklig solutio of the beam-colum subjected to ed momets are used to accurately capture the secod-order effects. The proposed method uses the force iterpolatio fuctios icludig a momet magificatio due to the axial force ad lateral displacemet. Thus, oly oe elemet per a physical member ca accout for the iteractio betwee the bedig momet ad the axial force i a ratioal way. To verify the accuracy ad computatioal efficiecy of the proposed method, the results are compared with those obtaied from the experimets, theoretical equatios, ad OpeSees (2008), which is a software framework for research i performace-based earthquake egieerig at the Pacific Earthquake Egieerig Research Ceter. The details of the proposed method are ow preseted. 2. Beam-colum elemet formulatio 2.1 The priciple of virtual work The priciple of virtual work states that exteral virtual work is simply the product of the displacemets ad their applied virtual forces, ad iteral virtual work is expressed by the product of strai resultats ad their virtual stress resultats itegrated over the cross-sectio. where δf( x) T d( x) δm( x) T ϕ x F d ( ) dx { θ 1z θ 2z θ 1y θ 2y d x } T { M 1z M 2z M 1y M 2y P} T ϕ M { φ z φ y ε x } T { M z M y P} T (1) (2) (3) (4) (5)

3 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 765 Fig. 1 Forces ad deformatio at the elemet ad sectio levels Fig. 1 shows that the elemet deformatios of Eq. (2) ad elemet ed forces of Eq. (3). Oe elemet has five degrees of freedom: a axial extesio ad two rotatios per each ode. Elemet forces of Eq. (3) idicate the correspodig axial force ad bedig momets. Fig. 1 also shows the sectio deformatios of Eq. (4) ad sectio forces of Eq. (5). Sectio deformatios are three strai resultats: a axial strai ad two curvatures with respect to local z ad y axes. Sectio forces iclude the compressive axial force ad two sectioal bedig momets. All of the fields give i Eqs. (2) to (5) are fuctios of the axial coordiate x, which is measured from the left ed of the beam, as show i Fig Displacemet-based ad flexibility-based formulatios I the displacemet-based formulatio (usually called the stiffess method), the displacemet fields are approximated by displacemet iterpolatio fuctios as ϕ = A(x)d (6) where A(x) cotais the derivatives of the displacemet iterpolatio fuctios as where A( x) N 1, xx 0 N 1, xx N 1, x v N 1, xx v N 1, xx θ N 2, xx θ N 2, xx v N 2, xx v θ N 2, xx θ 0 0 u u N 2, x N 1 v x2 v 1; N x x3 2 x x 3 x 3 x 2 θ N x2 θ --- x; N (7) (8) (9)

4 766 Jug-Woog Park ad Seug-Eock Kim u N 1 1 x (10) -- x u ; N 2 -- The mai shortcomig of the stiffess method is that the predefied displacemet iterpolatio fuctios do ot correspod to the exact solutio of the beam problem except for i special cases. Sice the assumptio of cubic iterpolatio fuctios gives a liear curvature distributio alog the elemet, a highly refied mesh is eeded to accurately capture the respose of the regios with highly oliear curvature distributio. Also, equilibrium is satisfied oly i a itegral sese over the elemet, but ot locally at each sectio alog the beam. The pricipal of virtual work ad Taylor s series expasio give a liear algebraic problem as follows F A T M R dx A T kadx d (11) where M R is the sectio resistig force vector, ad k is the sectio stiffess matrix. The left had side term i Eq. (11) is the ubalace force give as the differece betwee the exteral elemet force ad the elemet resistig force, ad is a fuctio of the deformed cofiguratio. The solutio of this oliear problem by Newto s method ivolves iteratively solvig the system of Eq. (11). I the force-based formulatio (usually called the flexibility method), the force fields are expressed as a fuctio of the elemet odal forces M = B(x)F (12) where B(x) is the force iterpolatio fuctios that eforce a liear bedig momet distributio alog the elemet ad a momet magificatio due to the axial force ad lateral displacemet: B( x) x x δ y x -- 1 x -- δ z (13) where δ y ad δ z are the lateral displacemets for the local y ad z axes, respectively. The terms of δ y ad δ z i Eq. (13) are ewly added oes i this study. Sice the curvature ca be approximated by the secod derivative of the lateral displacemet, δ y ad δ z are obtaied from solvig the differetial equatios as (Che ad ui 1987) δ y M 1z EI z k z cosk z sik z x sik z cosk z x x -- 1 M 2z EI z k z sik z x sik z x -- (14) M δ z cosk 1y y x 1 x sik y x cosk y x sik y x -- (15) 2 2 EIk y sik y EIk y sik y 2 2 where k z P EI z ad k y P EI y. Sice the sectio flexibility relates the strai resultats to the sectio momets, ad the elemet stiffess relates the elemet ed forces to the elemet ed displacemets, the strai resultats are give as ϕ = fb(x)kd (16) where f is the sectio flexibility matrix ad K is the elemet stiffess matrix. Comparig Eq. (6) ad Eq. (16) expresses the displacemet iterpolatio fuctio A(x) as M 2y

5 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 767 A(x) = fb(x)k (17) From Eq. (17), A(x) ad B(x) are depedet upo each other, ad this property is show i geeral flexibility-based formulatios. The mai advatage of the flexibility methods is that force iterpolatio fuctios iside the elemet are better suited to describe the oliear behavior of structural members. I the absece of elemet loads, a liear bedig momet distributio alog the elemet satisfies equilibrium i a strict sese. Accordigly, it provides the beefit of computig the exact elemet flexibility matrix. 2.3 Elemet stiffess accoutig for P δ effect To capture the effect of the axial force actig through the lateral displacemet of the beam-colum elemet relative to its chord (P δ effect), the slope-deflectio equatios for a beam-colum were preseted by Che ad ui (1987). Geerally oly oe elemet per a physical member ca accurately accout for the P δ effect. The elemet ed forces ad the elemet ed displacemets are related as M 1z M 2z M 1y M 2y P 1 -- EI z s zii EI z s zij θ1z EI z s zij EI z s ii θ 2z 0 0 EI y s yii EI y s yij 0 θ 1y 0 0 EI y s yij EI y s yii 0 θ 2y EA d where s zii, s zij, s yii ad s yij are the stability fuctios with respect to the local z ad y axes, ad are give as k s sik ( k ) 2 cosk ii (19a) 2 2cosk k sik ( k s ) 2 k sik ij (19b) 2 2cosk k sik where the subscript represets for z or y The three-dimesioal slope-deflectio equatios for a beam-colum that is ot subjected to trasverse loadigs ad relative joit traslatio ca be expressed i symbolic form as M = K e d (20) For members subjected to a axial force that is tesile rather tha compressive, the stability fuctios i Eq. (19) are redefied as s ii ( k ) 2 coshk k sihk cosk k sik (18) (21a) k s sihk ( k ) 2 ij (21b) 2 2coshk k sihk For a pied-eded perfectly straight colum subjected to a compressive axial force, the theoretical load-deflectio curve of the colum bifurcates ito stable ad ustable equilibrium braches at the poit whe s ii is equal to s ij.

6 768 Jug-Woog Park ad Seug-Eock Kim 2.4 Elemet stiffess accoutig for P effect The P effect is the effect of member forces actig through the relative trasverse displacemet of the member eds. If the member is permitted to sway, a additioal axial ad shear force will be iduced i the member. We ca relate this additioal axial ad shear force due to a member sway to the member ed displacemets as { } (22) { F } s [ K] s d where {F } s ad {d } are ed force ad displacemet vectors, ad {K} s is the elemet stiffess matrix give as (Kim et al. 2006) where ad [ K s ] [ K s ] [ K s ] [ K s ] T [ K s ] [ K s ] M 1z M 2z 2 0 a b 0 0 a c b 0 c M 1y M 2y 2 a ; b ; c P -- (23) (24) (25) Usig equilibrium ad kiematic relatios, the trasformatio matrix is give as [ T] R The total elemet stiffess matrix is ow give by [ K] t [ T R ] T [ K e ][ T R ] [ K] s (26) (27) The use of Eq. (27) requires iterative solutio techiques sice the sectio forces i a member chage durig the iteratio process. Excessive P effects will evetually itroduce sigularities ito the solutio, idicatig physical structural istability. Such behavior is clearly idicative of a poorly desiged structure that is i a eed of additioal stiffess. 2.5 Fiber model The resultat force ad momet ca be calculated by itegratig the tractios over the crosssectioal area, as show i Fig. 2 (Hjelmstad 1997),

7 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 769 Fig. 2 Tractio vector actig o a plae N( x) t x ( ) da M( x) p( zy, ) t x ( ) da (28) (29) where t(x) is tractio vector actig o a plae perpedicular to the logitudial axis ad p(z,y) is the positio vector of this tractio vector i the plae. Usig a oe-dimesioal versio of the Cauchy formula relatig stress to tractios, Eqs. (28) ad (29) ca be rewritte as N ( ε 0 yφ z zφ y )EdA M z y( ε 0 yφ z zφ y )EdA M y z( ε 0 yφ z zφ y )EdA I the fiber model, the elemet is subdivided ito a umber of logitudial fibers. The ceter coordiates i the local z y referece system, ad the fiber areas are used i formulatig the elemet stiffess matrix. The costitutive relatio of the sectio is ow expressed from the uiaxial respose of the fibers (30) (31) (32) k 2 E i A i y i E i A ( i y )z i E i i A i y i i 1 i 1 E i A ( i y )z i 2 E i i A i z i i 1 i 1 E i A i ( y i ) i 1 i 1 E i A i z i i 1 i 1 i 1 ( ) E i A i z i E i A i (33) The M ad ϕ are duals of each other, i the sese that they are meat to represet exactly the same costitutive relatio. I a flexibility-based method, the sectio deformatios is determied from the give sectio forces.

8 770 Jug-Woog Park ad Seug-Eock Kim 3. Costitutive relatioships Cocrete-filled steel tubes (CFT) are becomig icreasigly popular i recet decades due to their excellet performace such as high ductility ad improved stregth without icreasig the size of the colum. Also, usig CFT members makes the costructio easier by elimiatig the arragemet of formwork ad reiforcemet. Whe CFT members are subjected to compressive strais, both the steel tube ad the cocrete core experiece a lateral expasio. The lateral expasio of cocrete core gradually becomes greater tha the steel tube due to the chage of the Poisso ratio of the cocrete. At this stage, a radial pressure develops betwee the two media, ad the steel tube restraits the cocrete core to expad laterally. The effect of cofiemet o the cocrete core primarily depeds o the lateral pressure provided by the steel tube. Susatha et al. (2001) has performed extesive parametric aalyses to propose mathematical equatios for the average maximum lateral pressure, f rp i the box ad octagoal shaped CFT colums. For box shaped CFT colum, f rp is as follows: f rp 6.5R ( f c' ) ( f c ') 1.03 (34) b 12( 1 v 2 ) R (35) t 4π E 2 s where f rp is the lateral cofiig pressure, R is the width-to-thickess ratio parameter, b is the width of the sectio, t is the thickess of the steel tube, ad v is the Poisso ratio of the steel. The compressive stregth of cofied cocrete is give as f cc f c ' 4.0f rp βf c ' (36) where β is the stregth ehacemet factor. The Fig. 3 shows the stregth ehacemet factors with respect to Rf c ' ratios that deped o the geometry ad material properties. The compressive stress-strai curve for the cofied cocrete is defied for the pre-peak regio as Fig. 3 Stregth ehacemet factor

9 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 771 (Popovics 1973; Mader et al. 1988) ( ε ε f c f cc )r cc r 1 ( ε ε cc ) r (37) r E c E c ( f cc ε cc ) (38) ε cc ε f cc (39) f c ' where ε cc is the compressive strai correspodig to the peak stregth f cc. The slope, Z of the postpeak behavior proposed by Susatha et al. (2001) is expressed as ε cu Z 0 for R--- f c' R--- f c', for R--- f c' > for R--- f c' R--- f c' 2.4R f --- c' for R--- f c' < < f y for R--- f c' > (40) (41) Accordig to the Eqs. (40) ad (41), thick-walled steel tubes with low-stregth filled-i cocrete provide higher ductility capacity tha the thi-walled tubes with high-stregth filled-i cocrete. For low B/t ratios, local bucklig usually takes place i the post-peak regio of the load-deflectio curve of the colum (Tort ad Hajjar 2004). This type of respose is ductile ad it ca be esured by specifyig a maximum allowable B/t value accordig to the AISC Specificatio (2005). Cocrete stregth betwee cracks is geerally modeled to reduce gradually after a crack forms based o tesio stiffeig. I this study, a liear tesio softeig model is applied. The compressive ad tesile stress-strai relatioships of ucofied ad cofied cocrete are show i Fig. 4. The stress-strai relatioship for a reiforcemet bar is assumed to be tri-liear. It cosists of a iitial liear elastic regio, a yield plateau, ad a liear strai-hardeig phase. 4. Curret desig codes Desig methods for CFT colums are available i various major desig codes such as the AISC (2005), the ACI (2005), the Architectural Istitute of Japa (1997), the Europea Code EC 4 (2004). The desig methods of AISC ad ACI codes are briefly summarized below.

10 772 Jug-Woog Park ad Seug-Eock Kim Fig. 4 Stress-strai relatioships of cocrete 4.1 America Istitute of Steel Costructio (2005) The 2005 AISC (2005) ow uses a cross-sectioal stregth approach for colum desig cosistet with that used i reiforced cocrete desig (ACI 2005). The available axial stregth, icludig the effects of bucklig, ad the available flexural stregth ca be calculated usig either the plastic stress distributio method or the strai-compatibility method. The simplified approaches ca be applied to take advatage of stregth determiatio usig a limited umber of cases ad iterpolatio for all other cases o the poits of iteractio diagram. The omial compressive stregth of rectagular CFT colum is give as P P e P P for P e 0.44P 0 (42) where P 0.877P e for P e < 0.44P 0 (43) P 0 A s 0.85A c f c ' (44) P e π 2 EI eff K ( ) 2 where is the effective stiffess of composite sectio give as EI eff E s I s C 3 E c I c A s A c A s C The maximum B/t ratio for a rectagular CFT colum shall be equal to 2.26 E s. (45) (46) (47) 4.2 America Cocrete Istitute (2005) The desig cocept of a CFT colum is essetially the same as that of a ordiary reiforced

11 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 773 cocrete colum. To apply strai compatibility method, a cotiuous steel tube i a CFT is coverted ito equivalet reiforcig bars aroud the filled-i cocrete. It assumes that the cocrete has reached its crushig stregth i compressio at a strai of with a rectagular stress block. However, the ultimate stress is take as f c ' istead of 0.85f c ' to reflect that cocrete iside tubes does ot split with providig high ductility ad improved stregth. Slederess effect should be cosidered i the followig cases kl u > 22 r for o-sway case (48) kl u M 1 > for sway case (49) r M 2 where kl u is the effective legth, M 1 ad M 2 are smaller ad larger ed momets, respectively, ad r is a radius of gyratio. The momet amplified for the effects of member curvature is give as M c C m M 0 1 P ( 0.75P c ) (50) C m M M 2 (51) where C m is a correctio factor relatig the actual momet diagram to a equivalet uiform momet diagram, ad M 1 /M 2 is positive if the colum is bet i sigle curvature, ad egative if the member is bet i double curvature. The beam-colum aalysis usig strai compatibility method ca be readily implemeted i a spreadsheet. 5. Numerical examples To verify the accuracy of the proposed method, the 116 rectagular CFT colums tested to failure, five ed-restraied steel colums, ad a 3-dimesioal CFT frame structure are aalyzed. Failure stregths of the 116 rectagular CFT colums are predicted by the AISC, ACI , ad the proposed method, ad the results are compared with those obtaied from the experimets of the followig 6 ivestigatios: Tomii ad Sakio (1979), iu (2004), iu, Gho, ad Yua (2003), ue, iu, ad Ye (2007), iu (2005), ad iu (2006). The details of the test specimes ad stregth ratios (P test /P or M test /M ) are give for each group of test results i Tables 1-6 ad collectively i Fig. 5. A brief descriptio of each testig program from which the test results were extracted is preseted below. Tomii ad Sakio (1979) tested 36 rectagular CFT specimes which are composed of 8 cocetrically loaded colums ad 28 colums subjected to axial load ad bedig momets. After the costat axial load was applied, a mootoic icreasig momet was applied to the specimes. The rotatio, deflectios ad strais were recorded betwee load applicatios. The deflectios were held costat durig this time. This resulted i a decrease of bedig momet due to creep, but sice the readigs were performed relatively quickly, it was assumed that the creep effects o the measuremets were probably uimportat. They reported that the magitude of costat axial load ad B/t ratio had a sigificat effect o a ielastic behavior, especially o the descedig brach of the momet-curvature relatioships.

12 774 Jug-Woog Park ad Seug-Eock Kim Table 1 Aalysis results for test data 1 (Tomii ad Sakio 1979) Specime B, H t f P test M test P test /P or M test /M c ' (MPa) (MPa) (kn) (kn m) ACI 05 AISC 05 Authors I-A I-B II-A II-B III-A III-B IV-A IV-B I I I I I I I-6' II II II II II II II III III III III III III III IV IV IV IV IV IV IV Average Coefficiet of variatio

13 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 775 Table 2 Aalysis results for test data 2 (iu 2004) Specime B H t f c ' (MPa) (MPa) P test (kn) M test (kn m) P test /P or M test /M ACI 05 AISC 05 Authors E E E E E E E E E E E E Average Coefficiet of variatio Table 3 Aalysis results for test data 3 (iu, Gho, ad Yua 2003) Specime B H t f c ' (MPa) (MPa) P test (kn) M test (kn m) P test /P or M test /M ACI 05 AISC 05 Authors C C C C C C C C C C C C NA NA C C C C C C C C C C Average Coefficiet of variatio

14 776 Jug-Woog Park ad Seug-Eock Kim Table 4 Aalysis results for test data 4 (ue, iu, ad Ye 2007) Specime B H t f c ' (MPa) (MPa) P test (kn) M test (kn m) P test /P or M test /M ACI 05 AISC 05 Authors C C C C Average Coefficiet of variatio Table 5 Aalysis results for test data 5 (iu 2005) Specime B H t f c ' (MPa) (MPa) P test (kn) M test (kn m) P test /P or M test /M ACI 05 AISC 05 Authors R R R R R R R R R R R R R R R R R R R R R R Average Coefficiet of variatio iu (2004) performed the experimetal study o the behavior of 12 high stregth rectagular CFT colums subjected to eccetric loadig. The axial load was slowly applied to the specime by careful maipulatio of the loadig ad uloadig values. Durig the test, the logitudial ad the trasverse strais as well as the i-plae ad out-plae deflectios of the specime were recorded at a load icremet of 50 kn. The out-plae deflectio of the specime was less tha 1 mm, thus the

15 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 777 Table 6 Aalysis results for test data 6 (iu 2006) Specime B H t f c ' (MPa) (MPa) P test (kn) M test (kn m) ACI 05 P test /P or M test /M AISC 05 Authors S S S S S S S S S S S S S S S S Average Coefficiet of variatio specime was coformed to be uder compressio combied with uiaxial bedig. For specimes with slederess ratio of 20, crushig of cocrete ad local bucklig of steel tube were observed. For specimes with slederess ratio of 50, the failure loads were reached followed by local bucklig of steel tube at the mod height of the specime. iu, Gho, ad Yua (2003) ivestigated the ultimate capacity of 22 high-stregth rectagular CFT colums with cross-sectioal aspect ratio of 1.0, 1.5, ad 2.0. The mai parameters of the test specimes are the stregths of cocrete ad steel, cross-sectioal aspect ratio, ad volumetric steelto-cocrete ratio. The two horizotal flat plates welded at both eds of the specime were to esure that the steel hollow sectio ad the core cocrete were simultaeously loaded durig the test. I additio, local yieldig at both eds of the steel hollow sectio could be avoided. The specimes were tested to failure uder axial cocetric loadig. The failure mechaism was idetified as the material yieldig of steel hollow sectios ad the crushig of core cocrete. ue, iu, ad Ye (2007) tested twety four 1855 mm log rectagular CFT colums of mm. The specimes were divided ito four groups, ad the sectios i each group are filled with approximate cocrete stregth f c ' of 29, 63, 70, ad 84 MPa, respectively. Two bearig plates ( mm) were welded at the top ad bottom eds of each specime with eight spot-welded stiffeers to provide the rigidity plae at the eds of the specime whe the rotatio occurs at the oset of bucklig. Most of the specimes with ormal-stregth cocrete failed

16 778 Jug-Woog Park ad Seug-Eock Kim i global bucklig. Pure local bucklig cases were ot detected, ad the failure modes are either global bucklig or mixed global-local bucklig. iu (2005) tested 22 high-stregth rectagular CFT colums uder cocetric loadig. The test variables iclude the material stregths ( f c ' 60 ad 89 MPa), cross-sectioal aspect ratio ( ) ad volumetric steel-to-cocrete ratio ( ). Cocrete was the vertically cast ito the steel hollow sectio i three layers. Each layer of cocrete was compacted by a poker vibrator. The the cocrete was cured iside the steel hollow sectio with top ope to the air for two weeks util a 10- mm-thick flat plate was welded to the top to form a complete specime. It was reported that the ductility ehacemet was sigificat due to the cofiemet by the steel sectio. iu (2006) tested 16 short ad 4 sleder CFT colums uder eccetric loadig about major axis. The CFT specime was cured i the laboratory with top ope to the air for 14 days for the cocrete to set. High-stregth cemet mortar was subsequetly ifilled to flush the cocrete core with the steel tube. Fially, the top cap plate was welded to form a complete specime. It was observed that the short CFT colums with load eccetricity ratio with caot fully develop material plasticity at the failure load. Hece, plastic assumptios will ot be suitable for the umerical aalysis o them. The four sleder colums performed i a much similar failure mode to each other, ad the failure was characterized as overall bucklig. Fig. 5 shows that the predictios by the ACI ad AISC are geerally coservative with Fig. 5 Ratio of measured-to-calculated stregths by differet methods

17 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 779 mea values of 1.12 ad 1.17, respectively, ad coefficiets of variatio of 0.14 for both methods. For sleder colums, the predictios by the two code methods are more coservative ad scattered with mea values of 1.30 ad 1.43, ad COVs of 0.21 ad 0.18, respectively. The mai cause of this scatter may be the difficulty i evaluatig the geometric secod-order effects for sleder colums subjected to axial load ad bedig momets usig the simplified equatios i the code practices. The calculated capacities by the proposed method are sigificatly accurate with little scatter or treds for rectagular CFT colums over a wide rage i cocrete stregths, various combiatios of loadig, ad various width-to-thickess ratios (B/t) ad colum legths. It would be useful to expad this evaluatio to iclude eve more test data ad other code provisios such as Eurocode 4 ad Architectural Istitute of Japa code. The 102 specimes are short colums i which the slederess effects ca be eglected whe the ACI code is applied for colum desig. The other 15 specimes are sleder colums, therefore, the secod-order effects must be cosidered by icreasig the momet. For a sleder colum, bucklig may occur whe all fibers of the cross-sectio are still elastic. Thus, the Euler load will gover the load-carryig capacity of the colum. For a short colum, material yieldig of the fibers i the cross-sectio usually occurs before bucklig takes place, so the yieldig force of the sectio will gover the limit state of the colum. For a medium legth colum, some of the fibers of the crosssectio may yield while some fibers still remai elastic. I this case, the failure may be more accurately described as combied bucklig of the colum ad material yieldig of the sectio. To ivestigate the capability of the proposed method for capturig the elastic bucklig or critical load, five ed-restraied steel colums with differet ed coditios as show i Fig. 6 are aalyzed. The aalysis of steel colums cocetrates o the bucklig behavior as a result of geometric secod-order effects while the material is i the rage of liear elastic. The colums are subjected to small ed momets or lateral ed forces i additio to axial compressive force to iitiate the desired bucklig mode. The exact value of Euler load or critical load ca be obtaied from the effective legth factor that is depedet o the support coditio of the colum. The effective legth factors for the pied-eded, oe ed fixed ad oe ed free, oe ed fixed ad oe ed higed, oe ed fixed ad oe ed guided, ad oe ed higed ad oe ed guided colums are 1.0, 2.0, 0.7, 2.0, ad 1.0, respectively. From Fig. 7 to Fig. 11, the theoretical critical loads are compared collectively with the results Fig. 6 Ed-restraied beam-colums

18 780 Jug-Woog Park ad Seug-Eock Kim Fig. 7 Pied-eded colum Fig. 8 Oe ed fixed ad oe ed free colum obtaied from the proposed method ad OpeSees (2008). The bucklig load is ot give directly i the oliear fiite elemet aalysis, but rather a complete load-deformatio respose is obtaied from the two methods. Fig. 7(a) to Fig. 11(a) show the relatioships betwee compressive axial forces ad lateral displacemets accordig to the differet ed momets or lateral ed forces. The oliearbeamcolum elemet object based o a force-based formulatio with five itegratio poits alog the elemet is used i the OpeSees aalysis. The proposed method accurately calculates the elastic bucklig loads for the various beam-colums with differet ed coditios eve usig oe beam-colum elemet. However, the OpeSees overestimates the critical loads by 34% for oe ed higed ad oe ed fixed member, ad by 20% for all other cases. Fig. 7(b) to Fig. 11(b) show the load-displacemet resposes by the proposed method ad OpeSees with respect to the umber of elemets i OpeSees aalysis model. As the umber of elemets icreases, the OpeSees result becomes closer to those of the proposed method, ad usig eight beam-colum elemets i the aalysis by OpeSees gives very similar results to the proposed method. A three-dimesioal CFT portal frame as show i Fig. 12 is aalyzed by the proposed method

19 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 781 Fig. 9 Oe ed fixed ad oe ed higed colum Fig. 10 Oe ed fixed ad oe ed guided colum ad OpeSees. The sectio of each itegratio poit is divided by 100 rectagular fiber segmets. The frame structure is subjected to horizotal force at ode 10 to impose the torsio as well as gravity loads. It is assumed that steel tube ad cocrete are fully costraied, ad the local bucklig ca be avoided by usig the CFT sectio with B/t ratio withi the rage specified i the AISC specificatio (2005). The oliearbeamcolum elemet object based o a force-based formulatio with five itegratio poits alog the elemet is used, ad each member is modeled by te beam-colum elemets i the OpeSees. Fig. 13 shows the deformed shape obtaied from the proposed method i exaggerated scale. The load-displacemet resposes for the directio of x-axis at ode 5 ad ode 10 are show i Fig. 14 with respect to the umber of elemets. The respose for the directio of z-axis at ode 10 is show i Fig. 15. The resposes of the proposed method usig oe ad four elemets are i good agreemet with those obtaied from the four oliear beam-colum elemets of OpeSees, as show i Fig. 14 ad Fig. 15. The critical load calculated by usig oe oliear beam-colum elemet of OpeSees is about 10% higher tha that obtaied by four elemets of the same elemet object. The stresses of cocrete ad steel fibers still remai elastic. Accordig to the aalysis results, a good accuracy is obtaied by the proposed method with

20 782 Jug-Woog Park ad Seug-Eock Kim Fig. 11 Oe ed higed ad oe ed guided colum Fig dimesioal CFT frame structure reduced computatioal cost. Preset work ca be cosidered as a progressive cotributio for egieerig desig ad performace evaluatio (Che 2008). 6. Coclusios I this study, a flexibility-based fiite elemet method has bee developed for oliear ielastic aalysis of steel-cocrete composite frames i the cotext of a stadard fiite elemet aalysis program. From the results of this study, the followig coclusios ca be made: 1) ACI ad AISC give reasoably coservative estimates for short CFT colums. However, for sleder colums, the predictios by the two code methods become more

21 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 783 Fig. 13 Deformed shape of the 3-D frame Fig. 14 oad-displacemet relatioships for the directio of x-axis Fig. 15 oad-displacemet relatioships at ode 10 for the directio of z-axis

22 784 Jug-Woog Park ad Seug-Eock Kim coservative ad scattered due to the difficulty i evaluatig the geometric secod-order effects for sleder colums subjected to axial load ad bedig momets usig the simplified equatios i the code practices. 2) The proposed method ca accout for the iteractio betwee the bedig momet ad the axial force i a ratioal way usig the stability fuctios obtaied from the exact bucklig solutio of a beam-colum. Thus, usig oly oe elemet per a physical member provides most accurate stregth predictios for the 116 rectagular CFT colums tested to failure. 3) The proposed method accurately capture the elastic bucklig or critical loads for the five edrestraied beam-colums with differet ed coditios eve usig oe beam-colum elemet. The aalysis results usig oe elemet by the proposed method for the three-dimesioal CFT frame structure are i good agreemet with those obtaied from the four oliear beamcolum elemets of OpeSees. 4) Good accuracy ca be obtaied by the proposed method with reduced computatioal cost. The method ca provide valuable isight ito the desig ad behavior of CFT beam-colums. Ackowledgemets This paper is a part of the result from the Stadardizatio of Costructio Specificatios ad Desig Criteria based o Performace( 06~ 11), the Costructio & Trasportatio R&D Policy ad Ifrastructure Project. Refereces ACI (2005), Buildig code requiremets for reiforced cocrete (ACI ) ad Commetary (ACI 318R-05). America Cocrete Istitute. AIJ (1997), Recommedatios for desig ad costructio of cocrete filled steel tubular structures. Architectural Istitute of Japa. AISC (2005), Specificatio for structural steel buildigs. America Istitute of Steel Costructio. Alemdar, B.N. ad White, D.W. (2005), Displacemet, flexibility, ad mixed beam-colum fiite elemet formulatios for distributed plasticity aalysis, J. Struct. Eg. ASCE, 131(12), Ayoub, A. (2003), Mixed formulatio of oliear beam o foudatio elemets, Comput. Struct., 81(7), Ayoub, A. ad Filippou, F.C. (2000), Mixed formulatio of oliear steel-cocrete composite beam elemet, J Struct. Eg. ASCE, 126(3), Che, W.F. (2008), Structural egieerig: Seeig the big picture, KSCE J. Civil Eg., 12(1), Che, W.F. ad ui, E.M. (1987), Structural stability: theory ad implemetatio, Elsevier, New York. Ciampi, V. ad Carlesimo,. (1986), A oliear beam elemet for seismic aalysis of structures, Proc. 8th Euro. Cof. Earthq. Eg., isbo, Portugal, Eurocode 4, ENV (2004), Desig of composite steel ad cocrete structures, Part 1.1, Geeral rules ad rules for buildig, Commissio of Europea Commuities. Hjelmstad, K.D. (1997), Fudametals for structural mechaics, Pretice Hall. Hjelmstad, K.D. ad Taciroglu, E. (2002), Mixed methods ad flexibility approaches for oliear frame aalysis, J. Costr. Steel Res., 58(5-8), Kim, S.E., Uag, C.M., Choi, S.H., ad A, K.Y. (2006), Practical advaced aalysis of steel frames cosiderig lateral-torsioal bucklig, Thi Wall Struct., 44(7), iu, D. (2004), Behaviour of high stregth rectagular cocrete-filled steel hollow sectio colums uder

23 Noliear ielastic aalysis of steel-cocrete composite beam-colums usig the stability fuctios 785 eccetric loadig, Thi Wall Struct., 42(12), iu, D. (2005), Tests o high-stregth rectagular cocrete-filled steel hollow sectio stub colums, J. Costr. Steel Res., 61(7), iu, D. (2006), Behaviour of eccetrically loaded high-stregth rectagular cocrete-filled steel tubular colums, J. Costr. Steel Res., 62(8), iu, D., Gho, W.M., ad Yua, J. (2003), Ultimate capacity of high-stregth rectagular cocrete-filled steel hollow sectio stub colums, J. Costr. Steel Res., 59(12), ue, D.M., iu, J.., ad Ye, T. (2007), Experimetal study o rectagular CFT colums with high-stregth cocrete, J. Costr. Steel Res., 63(1), Mader, J.B., Priestly, J.N., ad Park, R. (1988), Theoretical stress strai model for cofied cocrete, J. Struct. Eg. ASCE, 114(8), OpeSees user commad-laguage maual (2008), Pacific Earthquake Egieerig Research Ceter, Uiv. of Califoria, Berkeley. Popovics, S. (1973), A umerical approach to the complete stress strai curves for cocrete, Cemet Cocrete Res., 3(5), Spacoe, E., Ciampi, V., ad Filippou, F.C. (1996), Mixed formulatio of oliear beam fiite elemet, Comput. Struct., 58(1), Spacoe, E., Filippou, F.C., ad Taucer, F.F. (1996), Fibre beam colum model for oliear aalysis of R/C frames: Part I. formulatio, Earthq. Eg. Struct. D, 25(7), Susatha, K.A.S., Ge, H., ad Usami, T. (2001), Uiaxial-stress-strai relatioship of cocrete cofied by various shaped steel tubes, Eg. Struct., 23(10), Taucer, F.F., Spacoe, E., ad Filippou, F.C. (1991), A fiber beam-colum elemet for seismic respose aalysis of reiforced cocrete structures, Report No. EERC 91-17, Earthquake Egieerig Research Ceter, Uiv. of Califoria, Berkeley. Tomii, M. ad Sakio, K. (1979), Experimetal studies o the ultimate momet of cocrete filled square steel tubular beam-colums, T. Arch. Ist. JPN, 275, Tort, C. ad Hajjar, J.F. (2004), Damage assessmet of rectagular cocrete-filled steel tubes for performacebased desig, Earthq. Spectra, 20(4),